Burst statistics as a criterion for imminent failure

0.56 MB·English

by  Srutarshi Pradhan

BURST STATISTICS AS A CRITERION FOR IMMINENT
FAILURE
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0 SrutarshiPradhan1,AlexHansen2 andPerC.Hemmer3
0
2
Department of Physics, Norwegian University of Science and Technology, N–7491 Trondheim,

n Norway,[email protected], [email protected], [email protected]
a
J
2 Abstract: The distribution of the magnitudesof damage avalanchesduring a failure process typically
1 follows a power law. When these avalanches are recorded close to the point at which the
systemfailscatastrophically,wefindthatthepowerlawhasanexponentwhichdiffersfrom

]
theonecharacterizingthesizedistributionofallavalanches.Wedemonstratethisanalytically
i
c
forbundlesofmanyfiberswithstatisticallydistributedbreakdownthresholdsfortheindivid-
s
- ualfibers. In this case the magnitudedistributionD(∆) for the avalanchesize ∆ followsa
rl powerlaw∆−ξ withξ = 3/2nearcompletefailure,andξ = 5/2elsewhere. Wealsostudy
t
m a network of electric fuses, and find numerically an exponent2.0 near breakdown, and 3.0
elsewhere. Weproposethatthiscrossoverinthesizedistributionmaybeusedasasignalfor
.
t imminentsystemfailure.
a
m
-
d Keywords: Failure,Fiberbundlemodel,Fusemodel,Burststatistics, Crossover
n
o
c
[ 1 INTRODUCTION

1
Catastrophicfailures[1,2,3,4,5]areabundantinnature: earthquakes,landslides,mine-collapses,
v
8 snow-avalanches etc. are well-known examples. A sudden catastrophic failure is a curse to hu-
6 man society due to the devastation it causes in terms of properties and human lives. Therefore a
2
fundamental challenge is todetect reliable precursors ofsuch catastrophic events. This isalso an
1
0 important issueinstrengthconsiderations ofmaterialsaswellasinconstruction engineering.
7
During the failure process of composite materials under external stress, avalanches of differ-
0
/ ent size are produced where an avalanche consists of simultaneous rupture of several elements.
t
a Such avalanches closely correspond tothe bursts of acoustic emissions [6,7]which areobserved
m
experimentally duringthefailure processofseveralmaterials. Ifonecounts allavalanches tillthe
- complete failure, their size distribution is typically a power law. However we observed recently
d
n thatifonerecordsavalanches notfortheentirefailure process, butwithinafiniteinterval, aclear
o crossover behavior [8, 9] is seen between two power laws, with a large exponent when the sys-
c
: tem isfarawayfrom thefailure point and amuchsmaller exponent for avalanches inthevicinity
v
of complete failure. Therefore such crossover behavior in the burst statistics can be taken as a
i
X
criterion for imminent failure. In this report we discuss the crossover behavior in two different
r fracture-breakdown models- fiber bundle models [10-18] which describe the failure ofcomposite
a
1
materialunderexternalloadandfusemodels[1,19,20]whichdemonstratethebreakdownofelec-
trical networks. Ananalytic derivation ofthecrossover behavior isgiven forfiberbundle models
andnumericalstudyconfirmssimilarbehaviorinafusemodel. Thusweclaimthatsuchcrossover
behaviorcanbeusedasasignalofimminentfailure. Arecentobservation[21]oftheexistenceof
a crossover behavior in the magnitude distribution of earthquakes within Japan has strengthened
thisclaim.
2 FIBER BUNDLE MODEL
OurfiberbundlemodelconsistsofN elasticandparallelfibers,clampedatbothends,withstatis-
ticallydistributed thresholds forbreakdownofindividual fibers(Figure1). Theindividual thresh-
olds x are assumed to be independent random variables drawn from the same cumulative distri-
i
bution functionP(x)andacorresponding densityfunction p(x):
x
Prob(x < x) = P(x) = p(u) du. (1)
i
Z0
δ
F
Figure1. AfiberbundleofN parallelfibersclampedatbothends. TheexternallyappliedforceF corre-
spondstoastretchingbyanamountδ.
Whenever a fiber experiences a force equal to or greater than strength threshold x , it breaks
i
immediatelyanddoesnotcontributetothestrengthofthebundlethereafter. Themaximalloadthe
bundlecanresistbeforecompletebreakdowniscalledthecriticalloadanditsvaluedependsupon
theprobability distribution ofthethresholds. Twopopular examplesofthreshold distributions are
theuniform distribution
x/x for0 x x
P(x) = r ≤ ≤ r (2)
1 forx > x ,
( r
andtheWeibulldistribution
P(x) = 1 exp( (x/x )κ). (3)
r
− −
Herex isareference threshold, andthedimensionless numberκistheWeibullindex(Figure2.)
r
2
(B)
(A)
p(x) p(x)
x x x x
r r
Figure2. Theuniformdistribution(A)andtheWeibulldistribution(B)withκ=5(solidline)andκ=10
(dottedline).
Fiberbundlemodelsdifferinthemechanismforhowtheextrastresscausedbyafiberfailureis
redistributed amongtheunbroken fibers. Thesimplest modelsaretheequal-load-sharing models,
in whichthe load previously carried byafailed fiberis shared equally by allthe remaining intact
fibers in the system. In the present article we study the statistics of burst avalanches for equal-
load-sharing models.
2.1 Burst statistics
Theburst distribution D(∆)isdefined as theexpected number ofbursts inwhich ∆fibers break
simultaneously when the bundle is stretched steadily until complete breakdown. Hemmer and
Hansenperformedadetailstatisticalanalysis[14]tofindtheburstdistributionforthisquasi-static
situation. Forabundle ofmanyfibers theycalculated the probability ofaburst ofsize ∆starting
atfiberk withthreshold valuex as
k
∆∆−1 x p(x ) x p(x ) ∆−1 x p(x )
k k k k k k
1 exp ∆ , (4)
∆! − Q(x ) Q(x ) × − Q(x )
(cid:20) k (cid:21)(cid:20) k (cid:21) (cid:20) k (cid:21)
where Q(x) = 1 P(x). Since a burst of size ∆ can occur at any point before complete break-
−
down, the above expression has to be integrated over all possible values of x , i.e, from 0 to x
k c
where x is the maximum amount of stretching beyond which the bundle collapses completely.
c
Therefore theburstdistribution isgivenby[14]
D(∆) ∆∆−1e−∆ xc
= p(x)r(x)[1 r(x)]∆−1exp[∆r(x)]dx, (5)
N ∆! −
Z0
where
xp(x) 1 d
r(x) = 1 = [xQ(x)]. (6)
− Q(x) Q(x) dx
3
Theintegration yieldstheasymptoticbehavior
−5
D(∆)/N ∆ 2, (7)
∝
whichisuniversalundermildrestrictions onthethresholddistributions [15]. Asacheck,wehave
done simulation experiments for different threshold distributions. Figure 3 show results for the
uniform threshold distribution (2)andtheWeibulldistribution (3)withindexκ = 5.
100 100
10-2 10-2
(A) (B)
10-4 10-4
D(∆) D(∆)
N 10-6 N 10-6
10-8 10-8
10-10 10-10
1 10 1∆00 1000 10000 1 10 1∆00 1000 10000
Figure3. TheburstdistributionD(∆)/N fortheuniformdistribution(A)andtheWeibulldistributionwith
index5(B).Thedottedlinesrepresentthepowerlawwithexponent 5/2.Bothfiguresarebasedon20000
−
samplesofbundleseachwithN =106fibers.
2.2 Crossoverbehaviornear failurepoint
100
10-1 x = 0.9 x
D(∆) 0 c
D(1) 10-2
10-3 x = 0
0
10-4
1 ∆ 10
Figure4.Thedistributionofburstsforthresholdsuniformlydistributedinaninterval(x0,xc),withx0 =0
andwithx0 =0.9xc. Thefigureisbasedon50000samples,eachwithN =106fibers.
When all the bursts are recorded for the entire failure process, we have seen that the burst
distribution D(∆) follows the asymptotic power law D ∆−5/2. If we just sample bursts that
∝
occur near criticality, a different behavior is seen. As an illustration we consider the uniform
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threshold distribution, and compare the complete burst distribution with what one gets when one
samples merely burst from breaking fibers in the threshold interval (0.9x ,x ). Figure 4 shows
c c
clearly thatinthelattercaseadifferentpowerlawisseen.
From Eq. 6 we see that r(x) vanishes at the point x . If we have a situation in which the
c
weakest fiber has its threshold x just a little below the critical value x , the contribution to the
0 c
integral in the expression (5) for the burst distribution will come from a small neighborhood of
x . Sincer(x)vanishesatx ,itissmallhere,andwemayinthisnarrowintervalapproximate the
c c
∆-dependent factorsin(5)asfollows
(1 r)∆e∆r = exp[∆(ln(1 r)+r)]
− −
= exp[ ∆(r2/2+ (r3))] exp ∆r(x)2/2 (8)
− O ≈ −
h i
Wealsohave
′
r(x) r (x )(x x ). (9)
c c
≈ −
Inserting everything intoEq.(5),weobtaintodominatingorder
D(∆) = ∆∆−1e−∆ xcp(x ) r′(x )(x x )e−∆r′(xc)2(x−xc)2/2 dx
c c c
N ∆! −
Zx0
= ∆∆−2e−∆p(xc) e−∆r′(xc)2(x−xc)2/2 xc
r′(xc) ∆! x0
| | h i
∆∆−2e−∆ p(x )
= c 1 e−∆/∆c , (10)
∆! r′(x ) −
c
| | h i
with
2
∆ = . (11)
c r′(x )2(x x )2
c c 0
−
ByuseoftheStirling approximation ∆! ∆∆e−∆√2π∆,theburst distribution (10)maybe
≃
writtenas
D(∆)
= C∆−5/2 1 e−∆/∆c , (12)
N −
(cid:16) (cid:17)
withanonzeroconstant
C = (2π)−1/2p(x )/ r′(x ) . (13)
c c
Wecanseefrom(12)thatthereisacrossover ataburst(cid:12)length(cid:12)around∆ :
(cid:12) (cid:12) c
D(∆) ∆−3/2 for∆ ∆
N ∝ ( ∆−5/2 for∆ ≪ ∆cc (14)
≫
We have thus shown the existence of a crossover from the generic asymptotic behavior D
∆−5/2 tothepowerlawD ∆−3/2 nearcriticality, i.e.,nearglobalbreakdown. Thecrossover∝is
∝
auniversalphenomenon, independent ofthethreshold distribution p(x).
5
104 ∆−5/2 10000
103 1000 ∆−5/2
D(∆)102 ∆−3/2 D(∆) 100
∆−3/2
101 10
100 1
10-1 0.1
1 10∆ 100 1 10∆ 100
Figure5. Thedistributionofburstsfortheuniformthresholddistribution(left)withx0 =0.80xcandfora
Weibulldistribution(right)withx0 =1(square)andx0 =1.7(circle).Boththefiguresarebasedon50000
sampleswith N = 106 fiberseach. Thestraightlinesrepresenttwodifferentpowerlaws, andthearrows
locatethecrossoverpoints∆ 12.5and∆ 14.6,respectively.
c c
≃ ≃
Fortheuniformdistribution ∆c = (1 x0/xc)−2/2,soforx0 = 0.8xc,wehave∆c = 12.5.
−
FortheWeibulldistributionP(x) = 1 exp( (x 1)10),where1 x ,wegetx = 1.72858
c
− − − ≤ ≤ ∞
and for x = 1.7, the crossover point will be at ∆ 14.6. Such crossover is clearly observed
0 c
≃
(Figure 5) near the expected values ∆ = ∆ = 12.5 and ∆ = ∆ = 14.6, respectively, for the
c c
abovedistributions.
106
104 ∆−5/2
)
∆
(D ∆−3/2
102
100
1 10∆ 100
Figure 6. The distribution of bursts for the uniform threshold distribution for a single fiber bundle with
N =107fibers.Resultswithx0 =0,i.e.,whenallavalanchesarerecorded,areshownassquaresanddata
foravalanchesnearthecriticalpoint(x0 =0.9xc)areshownbycircles.
The simulation results we have shown so far are based on averaging over a large number of
samples. Forapplications itisimportant thatcrossoversignalcanbeseenalsoinasinglesample.
We show in Figure 6 that equally clear crossover behavior is seen in a single fiber bundle when
N is large enough. Also, as a practical tool one must sample finite intervals (x , x ) during the
i f
fractureprocess. Thecrossoverwillbeobservedwhentheintervalisclosetothefailurepoint[9].
6
3 BURST STATISTICS IN THE FUSE MODEL
106
I 105
∆−3
104
∆) 103
(
V D ∆−2
102
101
100
1 ∆10 100
Figure7. Afusemodelandtheburstdistribution:Systemsizeis100×100andaveragesaretakenfor300samples.
Ontheaverage,catastrophicfailuresetsinafter2097fuseshaveblown.Thecirclesdenotetheburstdistribution
measuredthroughouttheentirebreakdownprocess.Thesquaresdenotetheburstdistributionbasedonbursts
appearingafterthefirst1000fuseshaveblown.Thetrianglesdenotetheburstdistributionafter2090fuseshave
blown.Thetwostraightlinesindicatepowerlawswithexponentsξ=3andξ=2,respectively.
Totestthecrossoverphenomenoninamorecomplexsituationthanforfiberbundles, wehave
studied burst distributions in the fuse model [1]. It consists of a lattice in which each bond is a
fuse, i.e., an ohmic resistor as long as the electric current it carries is below a threshold value. If
the threshold is exceeded, the fuse burns out irreversibly. The threshold t of each bond is drawn
from an uncorrelated distribution p(t). The lattice is placed between electrical bus bars and an
◦
increasing current ispassed through it. Thelattice isatwo-dimensional square oneplaced at45
with regards to the bus bars, and the Kirchhoff equations are solved numerically at each node
assuming thatallfuseshavethesameresistance.
Whenonerecordsallthebursts,thedistributionfollowsapowerlawD(∆) ∆−ξwithξ = 3,
∝
which is consistent with the value reported in recent studies [19, 20]. We show the histogram in
Figure 7. With a system size of 100 100, 2097 fuses blow on the average before catastrophic
×
failuresetsin. Whenmeasuringtheburstdistribution onlyafterthefirst2090fuseshaveblown,a
different power law is found, this time with ξ = 2. After 1000 blown fuses, on the other hand, ξ
remainsthesameasforthehistogram recording theentirefailureprocess (Figure7).
In Figure 8, we show the power dissipation E in the network as a function of the number of
blown fuses and as a function of the total current. The dissipation is given as the product of the
voltage drop across the network V times the total current that flows through it. The breakdown
process starts by following the lower curve, and follows the upper curve returning to the origin.
It is interesting to note the linearity of the unstable branch of this curve. In Figure 9, we record
the avalanche distribution for power dissipation, D (∆). Recording, as before, the avalanche
d
distribution throughout the entire process as well as recording only close to the point at which
the system catastrophically fails, result in two power laws, with exponents ξ = 2.7 and ξ = 1.9,
respectively. Itisinteresting tonotethatinthiscasethereisnotadifference ofunitybetween the
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two exponents. The power dissipation in the fuse model corresponds to the stored elastic energy
in a network of elastic elements. Hence, the power dissipation avalanche histogram would in the
mechanical system correspond tothereleased energy. Suchamechanical system could serveasa
simplemodelforearthquakes.
200 200
150 150
E100 E100
50 50
0 0
0 500 1000 1500 2000 2500 3000 0 5 10 15 20
k I
Figure8. PowerdissipationE asafunctionofthenumberofbrokenbonds(left)andasafunctionofthe
totalcurrentI flowinginthefusemodel(right).
TheGutenberg-Richter law [1, 2] relating the frequency of earthquakes with their magnitude
is essentially a measure of the elastic energy released in the earth’s crust, as the magnitude of an
earthquake isthelogarithmoftheelasticenergyreleased. Hence,thepowerdissipation avalanche
histogram D (∆) in the fuse model corresponds to the quantity that the Gutenberg-Richter law
d
addresses in seismology. Furthermore, the power law character of D (∆) is consistent with the
d
form of the Gutenberg-Richter law. It is then intriguing that there is a change in exponent ξ also
forthisquantity whenfailureisimminent.
106
104
∆)
D(d
102
100
1 10 100
∆
Figure 9. The powerdissipation avalanchehistogram D (∆) forthe fuse model. The slopes of the two
d
straight lines are 2.7 and 1.9, respectively. The circles show the histogram of avalanches recorded
− −
throughtheentireprocess,whereasthesquaresshowthehistogramrecordedafter2090fuseshaveblown.
8
4 CONCLUDING REMARKS
Establishing a signature of imminent failure is the principal objective in our study of different
breakdown phenomena. The same goal is of course central to earthquake prediction scheme. As
indifferentfailuresituations, burstscanberecordedfromoutside-withoutdisturbingtheongoing
failure process, burst statistics are much easily available and also contain reliable information of
the failure process. Therefore, any signature in burst statistics that can warn of imminent system
failure would be very useful in the sense of wide scope of applicability. The crossover behavior
in burst distributions, wefound in the fiber bundle models, is such a signature which signals that
catastrophic failure is imminent. Similar crossover behavior is also seen in the burst distribution
and energy distributions of the fuse model. Most important is that this crossover signal does not
hinge on observing rare events and is seen also in asingle system. Therefore, such signature has
a strong potential to be used as useful detection tool. It should be mentioned that most recently,
Kawamura[21]has observed achange inexponent values ofthe local magnitude distributions of
earthquakes inJapan,beforetheonsetofamainshock(Figure10). Thisobservationhasdefinitely
strengthened our claim of using crossover signals in burst statistics as a criterion for imminent
failure.
4.5
entire Japan 1985-1998
4
r =30km, m=4
max c
3.5
y
c slope = 0.60
n 3
e slope = 0.88
u
2.5
q
e
r 2
f

t
n 1.5 0-100 days
e
v 100-200 days
e 1
200-300 days
0.5 all period
0
2 3 4 5 6 7
m
Figure10. CrossoversignatureinthelocalmagnitudedistributionsofearthquakesinJapan. Theexponent
ofthedistributionduring100daysbeforeamainshockisabout0.60,muchsmallerthantheaveragevalue
0.88[21].
ACKNOWLEDGMENT
S. P. thanks the Research Council of Norway (NFR) for financial support through Grant No.
166720/V30.
9
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